KABUL—The Taliban’s new leader is wooing back some disaffected members as the insurgency wages a new offensive in strategic Helmand province, say those close to the group, defying U.S. and Afghan government efforts to undermine it.

Maulavi Haibatullah Akhundzada, in power for just over two months, has been reaching out to key Taliban figures sidelined by his predecessor, Mullah Akhtar Mansour, according to these people.